Roughly speaking, a connected graph \(H\)  has a pendant appearance in a graph \(G\)  if deleting a bridge from \(G\)  yields a component isomorphic to \(H\) , see Section REF  for a full definition.
Pendant appearances in \(R_n\)  have been well studied,
with numerous applications, for example concerning symmetries and vertex degrees, see [1]}, [2]} and for example [3]}, [4]}.
Here we give much more precise results on pendant appearances in \(R_n\) , presented in Section REF ; and we use them to deduce Theorem REF  and further results concerning the fragment of \(R_n\)  (the part of the graph not in the giant component), presented in Section REF . For these latter results, we assume only that \(\mathcal {A}\)  has a growth constant, and show that the asymptotic joint distribution of the components in the fragment of \(R_n\)  which are `free for \(\mathcal {A}\) ' is a Boltzmann Poisson distribution.
We also extend the idea of a pendant appearance to the more general one of a `vertex-pendant' appearance, see Section REF ; and in fact we use Theorem REF  on vertex-pendant appearances in \(R_n\)  to deduce the key result Theorem REF  on pendant appearances. This introductory Section  ends with a brief discussion in Subsection REF  of related work.
